Fractional Alexander polynomials for image denoising

Hamid A. Jalab a,n, Rabha W. Ibrahim b

a Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, Malaysiab Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

a r t i c l e i n f o

Article history:Received 11 December 2013Received in revised form4 April 2014Accepted 6 June 2014Available online 17 June 2014

Keywords:Fractional calculusFractional differential operatorFractional integral operatorFractional mask windowImage denoising

a b s t r a c t

Image denoising is an important task in image processing. The interest in using afractional mask window operator based on fractional calculus has grown for imagedenoising. This paper mainly introduces the concept of fractional calculus and proposes anew mathematical method in using fractional Alexander polynomials for image denoising.The structures of nn fractional mask windows on eight directions of this algorithm areconstructed. Finally, we measure the denoising performance by employing experimentsbased on visual perception and by using peak signal-to-noise ratios. The experimentsillustrate that the improvements achieved are compatible with other standard smoothingfilters.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Noise is any undesired signal that contaminates animage. Digital image acquisition is the primary processby which noise appears in digital images, converting anoptical image into a continuous electrical signal. Noise,arising from a variety of sources, is inherent to all electronicimage sensors and electronic components in the imageenvironment. The goal of image denoising methods is torecover the original image contaminated by noise. Removingnoise from the original signal remains an interesting topic forresearchers. Several methods have been proposed to removethe noise and recover the true image, with each approachhaving its advantages and limitations [1].

Image denoising refers to the process of recovering adigital image that has been contaminated by all kinds ofnoise, while preserving as much as possible the texturesand edges present in the image. Image denoising consid-ered as an important task in image segmentation, feature

extraction, and texture analysis. Traditionally, linear mod-els, such as the Gaussian filter, have been commonly usedto reduce noise. These methods perform well in the flatregions of images. However, their limitation is the inabilityto well-preserve the edges. The nonlinear model, however,can handle edges better than linear models. Anotherdenoising method known as neighborhood filtering pre-serves a pixel by obtaining the average of the values of itsneighbors [2]. Recently, many scholars have applied thetheory of fractional calculus to image processing. Frac-tional calculus and its applications are important in severaldiverse areas of mathematical, physical, and engineeringsciences. Fractional calculus generalizes ideas of the calculusof integrals and the derivatives of any arbitrary real orcomplex order. The advantages of fractional derivatives areobvious in modeling the mechanical and electrical propertiesof real materials, as well as in the description of properties ofgases, liquids, rocks, and in many other fields [3,4].

Studies on fractional calculus that involve differentoperators, such as RiemannLiouville, Erd'elyiKober,WeylRiesz, Caputo, and GrnwaldLetnikov operators,have evolved during the past 40 years, and have extendedin other fields. Fractional calculus in the field of imageprocessing has gotten considerable attention in image

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Signal Processing

http://dx.doi.org/10.1016/j.sigpro.2014.06.0040165-1684/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: 6 03 7967 2503.E-mail addresses: hamidjalab@um.edu.my (H.A. Jalab),

rabhaibrahim@yahoo.com (R.W. Ibrahim).

Signal Processing 107 (2015) 340354

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texture enhancement [57] and image denoising [811].All the results that are based on fractional calculusoperators showed that these methods are effective andreliable, and resulted in high levels of permanent immu-nity against different types of noise.

The fractional integral is extensively used in imagedenoising algorithms. Hu et al. [8,12] proposed a fractionalintegral denoising algorithm and the implementation of afractional integral filter using fractional integral maskwindows on eight directions based on fractional calculusRiemannLiouville definition. Simulation experimentsshowed the feasibility of the proposed fractional integraldenoising algorithm. Guo et al. [13] proposed an imagedenoising algorithm using fractional integral maskwindows based on the GrnwaldLetnikov definition offractional calculus. Grnwald and Letnikov achieved fine-tuning of image denoising by setting a smaller fractionalorder and controlled the effect of image denoising byiteration.

In our previous study [9], we proposed a novel digitalimage denoising algorithm based on the generalized Srivas-tavaOwa fractional integral operator. The results illustratedthat the proposed algorithm has a good upgrading of thedenoised image. For image texture enhancement, Jalab andIbrahim [5] proposed a texture enhancement technique usingthe fractional order SavitzkyGolay differentiator. This tech-nique computes the generalized fractional order derivative ofthe input image using the sliding weight window over theimage. Jalab and Ibrahim [6] proposed a texture enhancementtechnique for medical images using fractional differentialmask windows based on the SrivastavaOwa fractionaloperators. Pu et al. [14] proposed fractional differential maskwindows based on GrnwaldLetnikov and RiemannLiou-ville for multiscale texture enhancement using six fractionaldifferential masks. Experiments proved that the nonlinearlyenhancing complex texture in a smooth area by fractionaldifferential-based approach approves to be visibly better thanthat by traditional integral-based algorithms. Gao et al. [15]proposed image enhancement based on improved fractionaldifferentiation by piecewise quadratic interpolation equation.Experiments showed that for texture-rich digital image, thecapability of nonlinearly enhancing comprehensive texturedetails by improved fractional differentiation is obvious.

In this paper, we utilize the concept of fractionalcalculus for image denoising to generalize the Alexanderpolynomial in two approaches, namely, Alexander poly-nomialfractional differential (AFD) and Alexander poly-nomialfractional integral (AFI).

The Alexander polynomials advantage over the othertechniques is that these polynomials can be computed byutilizing a skein relation, which can be employed invarious topics in mathematics and physics, such as opera-tor algebras and statistical mechanics [16].

The structures of nn fractional mask windows ofthese algorithms are constructed. The denoising perfor-mance is measured by employing experiments based onthe visual perception and by using peak signal-to-noiseratio (PSNR). The remainder of this paper is organized asfollows. In Section 2, we introduce the generalized frac-tional differential and fractional integral of the Alexanderpolynomial. The construction of the fractional differential

mask windows, which is the new method proposed in thiswork, is presented in Section 3. The experimental resultsand the comparison with other studies are shown inSections 4 and 5, respectively. Finally, the conclusion ispresented in Section 6.

2. Alexander polynomial

The Alexander polynomial is a knot invariant created in1923 by J.W. Alexander, with integer coefficients correspond-ing to each knot type. The Alexander polynomial was the onlyknown knot polynomial until the Jones polynomial wasderived in 1984. The Alexander polynomial is the main toolused to discuss a pair of curves known as a Zariski pair. Thispair can be defined as follows: a couple of curves C1 and C2 ofequal degree is used to design a Zariski pair. If neighborhoodsexist, then TCi P2 (projective plane) of Ci; i 1;2 suchthat TC1; C1 and TC2;C2 are diffeomorphic, while thepairs P2;C1 and P2;C2 are not homeomorphic (topologi-cally not equivalent). Our aim is to construct two types ofmask windows utilizing the Alexander polynomial and itsgeneralization.

Definition 1. The Alexander polynomial is written as [17]

t d1

n 1ntn ; n 1; :::;d1 1

where n is positive integer and

nt texp 2nid

texp 2ni

d

:

From (1), we can conclude the following nt:1 11 t2

ffiffiffi3

pt1;

2 10 t2t1;3 9 t21;4 8 t2t1;5 7 t2

ffiffiffi3

pt1;

6 t12: 2

2.1. Fractional calculus

The idea of fractional calculus was proposed over 300years ago. Abel, in 1823, investigated the generalized tauto-chrone problem and, for the first time, applied fractionalcalculus techniques in a physical problem. Liouville subse-quently applied fractional calculus to problems in potentialtheory. Since that time, fractional calculus has captured theattention of many researchers in all areas of sciences [4].

This subsection deals with some preli